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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク functional equation ： ウィキペディア英語版 functional equation In mathematics, a functional equation〔 〕〔 〕〔 〕〔 〕 is any equation that specifies a function in implicit form.〔 〕 Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional equations they satisfy. The term ''functional equation'' usually refers to equations that cannot be simply reduced to algebraic equations. ==Examples== * The functional equation ::$$ f(s) = 2^s\pi^\sin\left(\frac\right)\Gamma(1-s)f(1-s) :is satisfied by the Riemann zeta function. The capital denotes the gamma function. * The gamma function is the unique solution of the following system of three equations: ::$f(x)=\backslash ,\backslash !$ ::$f(y)f\backslash left(y+\backslash frac\backslash right)=\backslash frac\}f(2y)$ ::$f(z)f(1-z)=\backslash ,\backslash !\backslash ,\backslash ,\backslash ,$       (Euler's reflection formula) * The functional equation ::$f\backslash left(\backslash right)\; =\; (cz+d)^k\; f(z)\backslash ,\backslash !$ :where are integers satisfying = 1, i.e. $$ \begin a & b\\c & d\end = 1, defines to be a modular form of order . * Miscellaneous examples, not necessarily involving standard or named functions: ::$f(x\; +\; y)\; =\; f(x)\; +\; f(y)\backslash ,\backslash !$ (Cauchy functional equation) Exponentiating, ::$f(x\; +\; y)\; =\; f(x)f(y),\; \backslash ,\backslash !$ satisfied by all exponential functions ::$f(xy)\; =\; f(x)\; +\; f(y)\backslash ,\backslash !$, satisfied by all logarithmic functions ::$f(xy)\; =\; f(x)\; f(y)\backslash ,\backslash !$, satisfied by all powers ::$f(x\; +\; y)\; +\; f(x\; -\; y)\; =\; 2(+\; f(y))\backslash ,\backslash !$ (quadratic equation or parallelogram law) ::$f((x\; +\; y)/2)\; =\; (f(x)\; +\; f(y))/2\backslash ,\backslash !$ (Jensen) ::$g(x\; +\; y)\; +\; g(x\; -\; y)\; =\; 2(g(y))\backslash ,\backslash !$ (d'Alembert) ::$f(h(x))\; =\; h(x\; +\; 1)\backslash ,\backslash !$ (Abel equation) ::$f(h(x))\; =\; cf(x)\backslash ,\backslash !$ (Schröder's equation). ::$f(h(x))\; =\; (f(x))^c\backslash ,\backslash !$ (Böttcher's equation). :: (Julia's equation). ::$\backslash omega(\; \backslash omega(x,u),v)=\backslash omega(x,u+v)$ (Translation equation) ::$f(x+y)\; =\; f(x)g(y)+f(y)g(x)\backslash ,\backslash !$ (sine addition formula). ::$g(x+y)\; =\; g(x)g(y)-f(y)f(x)\backslash ,\backslash !$ (cosine addition formula). ::$f(xy)\; =\; \backslash sum\; g\_l(x)\; h\_l(y)\backslash ,\backslash !$ (Levi-Civita). *A simple form of functional equation is a recurrence relation. This, formally speaking, involves an unspecified functions on integers and also shift operators. One such example of a recurrence relation is :: $a(n)\; =\; 3a(n-1)\; +\; 4a(n-2)\backslash ,\backslash !$ *The commutative and associative laws are functional equations. When the associative law is expressed in its familiar form, one lets some symbol between two variables represent a binary operation, ::$(a\backslash circ\; b)\backslash circ\; c\; =\; a\backslash circ\; (b\backslash circ\; c)~.$ But if we wrote ''ƒ''(''a'', ''b'') instead of ''a'' ○ ''b'' then the associative law would look more like what one conventionally thinks of as a functional equation, ::$f(f(a,\; b),c)\; =\; f(a,\; f(b,\; c)).\backslash ,\backslash !$ One feature that all of the examples listed above share in common is that, in each case, two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the identity function) are inside the argument of the unknown functions to be solved for. When it comes to asking for ''all'' solutions, it may be the case that conditions from mathematical analysis should be applied; for example, in the case of the ''Cauchy equation'' mentioned above, the solutions that are continuous functions are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using a Hamel basis for the real numbers as vector space over the rational numbers). The Bohr–Mollerup theorem is another well-known example. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「functional equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.